Does unit with when you play me linger a chain if it was played due to promising future

No. Units never linger on the Chain — only Spells do (Rule 359.3). A unit played from Promising Future finalizes and leaves the Chain the instant it's finalized, entering the board exhausted like any other unit (Rule 337.2, FAQ #11942).

What does go on the Chain is its "when you play me" trigger. That Play Effect is placed on the Chain as a Pending item immediately after the unit resolves, and it's finalized once Promising Future (and the other played cards) fully resolve — so opponents can react to the trigger, but never to the unit itself (FAQ #7044).